You need log tables or a calculator for an accurate answer. log(2)+log(5)=1, so if log(2)=0.3 approx then log5 is 1-0.3=0.7 approx. (We know this because 2^10=1024 which is approximately 10^3. So 2=10^(3/10)=10^0.3, making 0.3=log[10]2 approximately, using [] to show the base.) However, we don't know the log to base 10 of 23 or 45 without tables or a calculator.
23 lies between 16 (2^4) and 32 (2^5) so its log to base 2 is between 4 and 5. 45 is between 25 (5^2) and 125 (5^3), so its log to base 5 is between 2 and 3. Therefore the result is roughly 4.5+2.5=7.
Another way of estimating the log is to use a linear correlation. Take y=log[5](x). When x=5 y=1, and when x=25 y=2, and when x=125, y=3. Consider the number 45. It lies between 25 and 125. Consider a right-angled triangle with height 1 and base 100. The base represents the gap between 25 and 125 and the height represents the difference of 1 between the logs of 25 and 125 (2 and 3 respectively). The slope of the triangle is 1/100=0.01. The difference between 45 and 25 is 20, so this is one fifth the way between 25 and 125. Therefore the log is about a fifth of the distance between 2 and 3, making it about 2.2 (1/5=0.2). So log[5](45) is about 2.2.
Now consider y=log[2](x). 23 is between 16 and 32 so y is between 4 and 5. This time the slope of the triangle is 1/16. 23-16=7, so 23 is 7/16 along the triangle's base. 7/16 of the height is about 0.4, so we add that to 4 to get 4.4 and we have the approximate log[2](23). Add this to 2.2 and we get 6.6. This is another approximation to the answer without using a calculator or tables.
[Using a calculator the answer is about 6.8888. 6.6 is accurate to about 96%, which isn't bad. Our first guesstimate of 7 was over by less than 2%. An average of the two guesses gives us about 99% accuracy.]